In this dissertation, a systematic study of wavelet-based earlyvisionprocessing using different theoretical foundations hasbeen presented.The work applies the heat conduction theory andthe differentialgeometry theory, respectively, to the selectiveimage smoothing problemand the range edge detection problem. Inthe first topic, a new wavelet-based selective image smoothingscheme using an image flux conduction model is proposed.Basically, an image smoothing process can be simulated by a heatconduction process through mathematical formulation. But a heatconduction process borrowed from thermal physics may not beapplied to image smoothing directly. Therefore, a discrete imageflux conduction model has been developed by modifyingtheoriginal heat conduction model. For computing the image fluxvaluesmore accurately, a differential operator derived fromwavelet-based approximation is introduced into the developedmodel and a new wavelet-based discrete conduction equation isproposed. This proposed new equation is proved to satisfy theso-called maximum-minimum principle. The principle provides amore restrictive stability criterion than the von Neumannstability. When an iterative function system satisfies thisprinciple, the system will eventually converge.The second issuewhich have been discussed in this dissertation wasthe range edgedetection problem. In this work, differential geometry wasintroduced as a theoretical foundation. A new 3-D invariantfeature,the magnitude of normal changes alone two orthogonaldirections, was introduced to detect the boundaries of differentsurface patches. Since the image data were discrete by nature,several surface approximation schemes including quadraticsurface fitting, orthogonal wavelet-based approach, and non-orthogonal wavelet-based approach were applied to approximatethe continuous data surface. Thus, the multiscale wavelettransform was introduced to detect the set of edge points. Thesepoints were selected once their magnitudes of a normal changeexceed a preset threshold. Experimental results havedemonstrated that the proposed wavelet-based approach is indeedsuperb in range edge detection.